A yoledet does not say "asher yatzar" after the delivery. "Asher yatzar" applies specifically to the bodily systems that on a daily level take in what is needed and release what is not, in order to ensure our survival. Giving birth is a unique event, not part of the body's regular daily functioning.

In practice, most women do urinate and defecate over the course of delivery and should make an "asher yatzar" over that when they are cleaned up, as long as they have not yet needed to urinate or defecate again.

One should not hesitate to prepare a personal prayer to say during or right after giving birth.